Stochastic Line Integrals Research Articles

Non-degeneracy of stochastic line integrals

Non-degeneracy of stochastic line integrals

Stochastic line integrals and stream functions as metrics of irreversibility and heat transfer

Stochastic line integrals are presented as a useful metric for quantitatively characterizing irreversibility and detailed balance violation in noise-driven dynamical systems. A particular realization is the stochastic area, recently studied in coupled electrical circuits. Here we provide a general framework for understanding properties of stochastic line integrals and clarify their implementation for experiments and simulations. For two-dimensional systems, stochastic line integrals can be expressed in terms of a stream function, the sign of which determines the orientation of nonequilibrium steady-state probability currents. Theoretical results are supported by numerical studies of an overdamped two-dimensional mass-spring system driven out of equilibrium. Additionally, the stream function permits analytical understanding of the scaling dependence of stochastic area growth rate on key parameters such as the noise strength for both linear and nonlinear springs.

A construction of diffusion processes associated with sub-Laplacian on CR manifolds and its applications

A diffusion process associated with the real sub-Laplacian $\Delta_b$, the real part of the complex Kohn–Spencer Laplacian $\square_b$, on a strictly pseudoconvex CR manifold is constructed via the Eells–Elworthy–Malliavin method by taking advantage of the metric connection due to Tanaka and Webster. Using the diffusion process and the Malliavin calculus, the heat kernel and the Dirichlet problem for $\Delta_b$ are studied in a probabilistic manner. Moreover, distributions of stochastic line integrals along the diffusion process will be investigated.

A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

In this paper we prove a Feynman–Kac–Ito formula for magnetic Schrodinger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrodinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman–Kac–Ito formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman–Kac–Ito formula, we prove a Kato inequality, a Golden–Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials.

Phase-space path integral and Brownian motion

We formulate and examine a phase-space path integral representation of a quantum time evolution in terms of Fock–Bargmann spaces and stochastic line integrals. The Fock–Bargmann space \documentclass[12pt]{minimal}\begin{document}${{\cal H}_{\rm FB}}$\end{document}H FB is a closed subspace of \documentclass[12pt]{minimal}\begin{document}$L^2({\cal M})$\end{document}L2(M), where \documentclass[12pt]{minimal}\begin{document}${\cal M}$\end{document}M is the phase space. Let EFB be the orthogonal projection onto \documentclass[12pt]{minimal}\begin{document}${{\cal H}_{\rm FB}}$\end{document}H FB , and h the classical Hamiltonian, then H = EFBhEFB is understood as a quantization of h. The quantum time evolution e−itH is asymptotically represented by the solution of a variant of diffusion equation with the imaginary potential ih. Thus the quantum time evolution can be represented stochastically, in terms of Brownian motions and stochastic integrals.

Large deviation for stochastic line integrals as L p -currents

The large deviation principle for stochastic line integrals along Brownian paths on a compact Riemannian manifold is studied. We regard them as a random map on a Sobolev space of 1-forms. We show that the differentiability order of the Sobolev space can be chosen to be almost independent of the dimension of the underlying space by assigning higher integrability on 1-forms. The large deviation is formulated for the joint distribution of stochastic line integrals and the empirical distribution of a Brownian path. As the result, the rate function is given explicitly.

Laplace approximation for stochastic line integrals

We prove a precision of large deviation principle for current-valued processes such as shown in Bolthausen et al. (Ann Probab 23(1):236–267, 1995) for mean empirical measures. The class of processes we consider is determined by the martingale part of stochastic line integrals of 1-forms on a compact Riemannian manifold. For the pair of the current-valued process and mean empirical measures, we give an asymptotic evaluation of a nonlinear Laplace transform under a nondegeneracy assumption on the Hessian of the exponent at equilibrium states. As a direct consequence, our result implies the Laplace approximation for stochastic line integrals or periodic diffusions. In particular, we recover a result in Bolthausen et al. (Ann Probab 23(1):236–267, 1995) in our framework.

Various types of stochastic integrals with respect to fractional Brownian sheet and their applications

In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let { B z H , z ∈ [ 0 , 1 ] 2 } be a fractional Brownian sheet with Hurst parameters H = ( H 1 , H 2 ) , and ( [ 0 , 1 ] 2 , B ( [ 0 , 1 ] 2 ) , μ ) a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in [ 0 , 1 ] 2 , and four types of stochastic surface integrals: ∫ φ ( s ) d B i γ ( s ) , i = 1 , 2 , ∫ α ( a ) d B a H , ∫ ∫ β ( a , b ) d B a H d B b H , ∫ ∫ β ( a , b ) d μ ( a ) d B b H , ∫ ∫ β ( a , b ) d B a H d μ ( b ) . As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters H 1 , H 2 ∈ ( 1 / 4 , 1 ) . Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process.

On large deviations for random currents induced from stochastic line integrals

On large deviations for random currents induced from stochastic line integrals

Sample path large deviations for a class of random currents

We study long-time asymptotic behavior of the current-valued processes on compact Riemannian manifolds determined by the stochastic line integrals. Sample path large deviation estimates are proved, which induce the law of the iterated logarithm as a corollary. As their application, we give a probabilistic approach to the analysis on noncompact Abelian covering manifolds.

Sample path large deviations for a class of random currents

We study long-time asymptotic behavior of the current-valued processes on compact Riemannian manifolds determined by the stochastic line integrals. Sample path large deviation estimates are proved, which induce the law of the iterated logarithm as a corollary. As their application, we give a probabilistic approach to the analysis on noncompact Abelian covering manifolds.

Analytic Functions on Abstract Wiener Spaces

Let (X, H, μ) be a real abstract Wiener space. A new definition of analytic functions on X is introduced, and it is shown that stochastic line integrals of real analytic 1-forms along Brownian motion and solutions to stochastic differential equations with real analytic coefficients are analytic Wiener functionals. An Lp-theoretical sufficient condition for Wiener functionals to be analytic and an associated Cauchy formula are also established.

Local Time and Stochastic Area Integrals

If $(B_t)_{t\geq 0}$ is Brownian motion on $\mathbb{R},$ if $A(t, x) \equiv \int^t_0I_{\{B_s\leq x\}} ds$ and if $\tau(\cdot, x)$ is the right-continuous inverse to $A(\cdot, x)$, then the process $\tilde{B}(t, x) \equiv B(\tau(t, x))$ is a reflecting Brownian motion in $(-\infty, x\rbrack$. If $\mathscr{E}_x$ denotes the $\sigma$-field generated by $\tilde{B}(\cdot, x)$, then $(\mathscr{E}_x)_{x\in\mathbb{R}}$ forms a filtration. It has been proved recently that all $(\mathscr{E}_x)$-martingales are continuous, in common with the martingales on the Brownian filtration. Here we shall prove that, as with the Brownian filtration, all $(\mathscr{E}_x)$-martingales can be written as stochastic area integrals with respect to local time. This requires a theory of such integrals to be developed; the first version of this was given by Walsh some years ago, but we consider the account presented here to be definitive. We apply this theory to an investigation of stochastic line integrals of local time along curves which need not be adapted processes and illustrate these constructs by identifying the compensator of the supermartingale $(L(\tau(t, x), x))_{x\geq a}$ previously studied by McGill.